Elementary Math in Elementary School: The Efect Of Interference On Learning The Multiplication Table Part 1

Abstract

Memorizing the multiplication table is a major challenge for elementary school students: there are many facts to memorize, and they are often similar to each other, which creates interference in memory. Here, we examined whether learning would improve if the degree of interference is reduced, and which memory processes are responsible for this improvement. 

Memory is a very important part of human life. It is not only related to our study and work but also closely related to the quality of our lives. However, over time, our memory may become blurry, which may cause a lot of inconvenience and distress in our lives. Today, we will discuss the relationship between memory interference and memory, and provide some positive suggestions to help everyone enhance their memory.

First, we need to understand why memory interferes. Memory interference refers to the interference of some information or experience that affects our memory process. Sometimes, when we accept new information, we will be affected by old information and produce wrong memories. This situation is called active interference. In addition, there is an effect called negative interference, which is when we try to recall something, our memory is interfered with by some irrelevant information and becomes unclear or wrong.

So, how to prevent memory interference? Here are some suggestions:

First, avoid distractions while studying. Try to study in a quiet environment and avoid noisy environments such as TV or music. It’s best not to bring your cell phone or other distracting devices with you while studying.

Second, try to consolidate the memory. Memory consolidation refers to maintaining the validity and reliability of memory information for a long time and storing it in a long-term memory bank. Methods to consolidate memory include repeated review, association with other experiences, and attempts to repeat information.

Third, use memory-enhancing techniques. Memory enhancement techniques include the use of association techniques, image techniques, memory palaces, and other methods to help memories be quickly and efficiently transformed into long-term memory banks.

Fourth, maintain an optimistic attitude. Negative emotions can interfere with our memory, so we should try to maintain positive emotions and an optimistic attitude as much as possible. Maintaining a happy attitude by chatting with friends and family, exercising, etc. can help enhance our memory.

In summary, memory interference has an impact on our memory, but by trying some preventive measures and memory enhancement techniques, we can better protect our brains and improve our memory. I hope everyone will pay attention to their memory and health in daily life and enjoy a better and more fulfilling life. It can be seen that we need to improve our memory. Cistanche deserticola can significantly improve memory because Cistanche deserticola is a traditional Chinese medicinal material with many unique effects, one of which is to improve memory. The efficacy of minced meat comes from the various active ingredients it contains, including acid, polysaccharides, flavonoids, etc. These ingredients can promote brain health in a variety of ways.

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In a series of 16 short training sessions over 4 weeks first-graded children learned 16 multiplication facts—4 facts per week. In 2 weeks the facts were dissimilar from each other (low interference), and in 2 control weeks, ks the facts were similar (high interference). Learning in the low-similarity, low-interference weeks was better than in the high-similarity weeks. Critically, this similar outcome originated in a specific learning context, i.e., the grouping of factintoto weeks, and could not be explained as an intrinsic advantage of certain facts over others. 

Moreover, the interference arose from the similarity between facts in a given week, not from the similarity to previously learned facts. Similarity affected long-term memory—its effect persisted 7 weeks after training had ended; and it operated on long-term memory directly, not via the mediation of working memory. Pedagogically, the effectiveness of the low interference training method, which is dramatically different from currently used pedagogical methods, may pave the way to enhancing how we teach the multiplication table in school.

Keywords:

Multiplication table, Proactive interference, Long-term memory, Math teaching methods.

Introduction

Learning the basic arithmetic facts, in particular, the multiplication table is a key part of the elementary school mathematics curriculum. Mastering the multiplication table is important not only in itself but also for acquiring more advanced mathematical skills: even if memorized multiplication facts can be solved by various workarounds (strategies, external devices), automatic knowledge is still advantageous because it can free cognitive resources that can be used for other tasks (Bratina & Krudwig, 2003; Hasselbring, 1988). 

Sadly, learning the multiplication table is not only important but, it is also difficult. Children typically learn by heart the single-digit multiplication facts from 3×3 to 9×9—a challenging quantity of 28 facts to remember. Other single-digit multiplications are not necessarily learned by heart, as they can be solved using rules, N×0, N×1, N×10; using twin addition, N×2; or using multi-stage procedures to solve multi-digit multiplication. 

Given this large memorization challenge, it is perhaps not surprising that many children have difficulties learning the multiplication table and show poor/abnormal performance patterns (Geary, 2004; Gross-Tsur et al., 1996; Noël & De Visscher, 2018; Räsänen & Ahonen, 1995).

Learning arithmetic facts, specifically the multiplication table has at least two aspects. One aspect pertains to the mathematical meaning of arithmetic facts. The mathematical meaning determines the result of each given fact, and it has several consequences—for example, that addition facts are related to counting and to the idea of moving along a number line; that a multiplication fact is equivalent to a series of same-operand additions; and that for both additions and multiplications, larger operands are correlated with larger results. 

Learning such mathematical truths is critical to understanding the meaning of arithmetic and being able to use it properly. It can also help compute the result of arithmetic facts—e.g. if we need to solve a problem whose solution we did not learn yet or we forgot. Moreover, several of these truths are not just mathematical, they may also affect the cognitive processing of the arithmetic facts. 

For example, the magnitude of arithmetic affects the difficulty of solving it (Groen & Parkman, 1972; Zbrodof & Logan, 2005), and solving addition and subtraction facts is associated with the activation of number-line representations (McCrink et al., 2007; Pinheiro-Chagas et al., 2017).

The present study focuses on the second aspect of the knowledge of arithmetic facts, in particular multiplication facts—rote memoryThisis aspect is extremely important too: although arithmetic facts have mathematical meaning, and understanding this meaning is a critical stage of learning them, most educated adults eventually come to learn most single-digit arithmetic facts by heart, and they solve them by retrieving a memorized response, and not (at least not only) by applying mathematical rules (Campbell & Beech, 2014). 

More specifically, multiplication facts are stored in verbal memory (Dehaene, 1992; Dehaene & Cohen, 1995; Dehaene et al., 2003). In line with this idea, learning multiplication facts depends on language skills and memory (LeFevre et al., 2010; Xu et al., 2021; this is true also for arithmetic facts other than multiplication); and phonological skills predict arithmetic abilities (Jordan et al., 2010; Korpipää et al., 2020; Simmons & Singleton, 2008). 

Thus, while understanding the mathematical aspects of multiplication is necessary— it provides a way to solve multiplication exercises, and it underlies the knowledge of how to use multiplication for particular goals—rote memory helps become proficient in arithmetic. Indeed, in several countries, typical multiplication lessons include not only conceptual learning but also rote memorization of the multiplication table using various strategies, e.g., recitation or songs (Olfos & Isoda, 2021).

Because multiplication facts are stored as individual facts in memory, they are subject to the limits of human memory, in particular to the interference induced by the similarity between items: memorizing similar items is hard because they interfere with each other similarieffectect is evident when learning arithmetic facts (Barrouillet t  al., 1997; Campbell & Graham, 1985; De Visscher & Noël, 2013; Katzof et al., 2020; Noël & De Visscher, 2018) as well as in other memory tasks (Baddeley, 1966a, 2003; Hall, 1971; Nelson et al., 19 74; Oberauer & Kliegl, 2006; Oberauer & Lange, 2008; Stager & Werker, 1997; Vallar, 2006)Specificallylly for arithmetic facts, one explanation of the similarieffectect is that the facts are represented in memory as a network of associations, in which each fact is associated not only with the correct solution but also with incorrect solutions (Campbell & Graham, 1985). The similarity between facts increases the likelihood of following these incorrect-solution associations and retrieving an incorrect answer. 

Some individuals have particularly high sensitivity to similarity-induced interference (“hyper-sensitivity to interference”), and consequently, they find it extremely hard to learn the multiplication table (De Visscher & Noël, 2013, 2014a; De Visscher et al., 2018; Dotan & Friedmann, 2019), i.e., they show symptoms of dyscalculia (American Psychiatric Association, 2013; World Health Organization, 1992).

As a means to overcome the difficulty caused by the similarity between facts, we propose a simple teaching method based on two foundations. firstfrst foundation is similarity versus dissimilarity: similar multiplication facts are hard to memorize because they interfere with each other, but it may still be easy enough to memorize a set of dissimilar multiplication facts (Campbell, 1987; De Visscher & Noël, 2013, 2014a, 2014b; Girelli et al., 1996; Katzof et al., 2020). Thus, learning the full multiplication table is hard, but it should be possible to learn a subset of multiplication facts as long as they are dissimilar from each other (e.g., 9×9=63 and 7×4=28). 

The second foundation is temporal distance: similar facts interfere with each other when they are presented simultaneously, or within a short time from each other, but interference should be lower when the facts are presented with sufficient temporal delay between them (Campbell, 1987). Thus, we may be able to teach similar facts (e.g., 8×8=64 and 8×6=48) if we present them with sufficient temporal delay between each other. These two foundations lead to the following simple teaching method: each lesson includes only dissimilar facts, and indifferent lessons that include similar facts are administered with sufficient temporal delay between them.

We examined whether this low-similarity training method would improve the learning of multiplication facts by first-grade children who had not yet started learning the multiplication table. the specific experimental design was as follows: during four weeks of training, each child learned four multiplication facts per week (16 facts overall). In two weeks, the 4 facts were dissimilar from each other. As a control, in the two other wee, ks the facts were similar to each other. Thus, each child learned both similar and dissimilar facts. We predicted better learning in the weeks with dissimilar facts than in the control weeks. As we shall see, this was indeed the case.

In a previous single-case study (Dotan & Friedmann, 2019), this low-similarity training method was extremely successful for a woman with hypersensitivity to interference. A small effort of about 4  min per multiplication fact, distributed over 4  weeks, sufficed for her to learn 12 facts and to remember them two months later with 80% accuracy—much better than her performance in a high-similarity control condition. 

However, Dotan and Friedmann’s study had several limitations, which limit its ability to inform reliably about learning the multiplication table in common educational settings. First, the study examined only one participant, so its pedagogical conclusions are suggestive at best. Second, the participant was an adult woman, so the study does not inform directly about the memorization processes in children— the population that attends school and learns the multiplication table. Third, this woman had a very specific memory disorder (hypersensitivity to interference), so her performance pattern may be different from that of individuals without cognitive disorders, or individuals with other types of cognitive disorders. Last, although Dotan and Friedmann showed that low-similarity training can improve memorization, they did not examine in detail why this improvement occurred. For example, they did not identify the specific memory mechanism responsible for the improvement.

The present study aimed to overcome these limitations. We had three goals, which have an impact on pedagogy as well as on cognitive theory. First, we aimed to find a simple way to make it easier to learn the multiplication table for the most relevant population—typically developing children in early elementary school grades. Second, we aimed to show, for the first time, causal evidence for the effect of similarity between multiplication facts on their memorizatiospecificallylly in typically developing elementary school children. Third, we asked why similarity disrupts memorization; in particular, we aimed to identify the specific memory mechanism responsible for the similarity effect.

Method

Participants

The participants were native Hebrew speakers and were recruited via social networks. The inclusion criteria were that the child: (1) had no reported or suspected learning disorders, and (2) did not yet learn the multiplication table or the meaning of multiplication—neither before the study nor during the 12 weeks of the study.

35 children started the study. Additionfilefle 1: Table S1 shows their detailsTheirir ages were between 6;1 (6 years, 1  month) and 7;11 (mean=7;1, SD=0;5). They are in the first grade (27 children), 2nd grade (7 children), or the last year of kindergarten (1 child). Of these, 18 children were excluded. Tree children were excluded immediately after the pre-experiment test because this test showed that they already knew some of the multiplication facts to be learned. 

Additional 15 children dropped along the way: 3 decided to quit; one was excluded for not following the study rules, which prohibited parental help; and 11 were excluded for being uncooperative or inattentive (for detailed exclusion reasons, and additional explanations, see Additionafile   1:  Table  S2). Importantly, because we used a within-participant design, and each child performed both experimental conditions (low-similarity training and high-similarity training), the participant exclusions were not confounded with the experimental manipulation. 

Furthermore, the specific exclusion reasons were not related to the magnitude of a putative similariteffectct: when we excluded a child based on objective measures, we never relied on a measure related to similarity, or on any performance measure reported in the results below; and when we excluded a child based on an experimenter’s impression of the child’s behavior in particular experiment sessions, the experimenters who made the decision were not told in advance which experimental condition was administered in those sessions.

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